AbstractThe purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant $$\Lambda <0$$
Λ
<
0
and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations $$\psi $$
ψ
of Kerr–AdS solving $$\Box _g\psi -\frac{2}{3}\Lambda \psi =0$$
□
g
ψ
-
2
3
Λ
ψ
=
0
with reflecting boundary conditions imposed at infinity. Understanding the behavior of $$\psi $$
ψ
at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass $${\mathfrak {m}} = M \sqrt{-\Lambda }$$
m
=
M
-
Λ
and angular momentum $${\mathfrak {a}} = a \sqrt{-\Lambda }$$
a
=
a
-
Λ
satisfy a certain non-Diophantine condition, then perturbations $$\psi $$
ψ
arising from generic smooth initial data blow up $$|\psi |\rightarrow +\infty $$
|
ψ
|
→
+
∞
at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for $$\Lambda \ge 0$$
Λ
≥
0
—in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters $${\mathfrak {m}}, {\mathfrak {a}}$$
m
,
a
for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking–Reall bound. On the other hand, we conjecture that for a set of parameters $${\mathfrak {m}}, {\mathfrak {a}} $$
m
,
a
which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon $$|\psi |\le C$$
|
ψ
|
≤
C
. This suggests that the validity of the $$C^0$$
C
0
-formulation of Strong Cosmic Censorship for $$\Lambda <0$$
Λ
<
0
may change in a spectacular way according to the notion of genericity imposed.
Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables
